Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

Abstract

This paper presents two exact explicit solutions for the three dimensional dual-phase lag (DLP) heat conduction equation, during the derivation of which the method of trial and error and the authors' previous experiences are utilized. To the authors' knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

“…Two exact explicit solutions for the 3-D DPL heat conduction equation are presented for benchmark purposes [141]. During the derivation of solutions, the method of trial and error is utilized and some special assumptions are adopted during the derivation to accomplish the variables separation.…”

“…Bioheat application [133,134] Laplace transformation technique [71,92,93,135,136] Cartesian, Cylindrical, Spherical 1-D, 2-D CV, DPL Hybrid method [93]. Moving heating source [136] Other analytical methods [137][138][139][140][141][142][143][144][145][146][147] Cartesian 1-D, 2-D, 3-D CV, DPL Fourier series [137]: Constant heat flux and constant temperature BC, [138,139]: fluctuating harmonic heating source. Spectral [140]: CV model, Chebyshev polynomials.…”

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

“…In addition, they are of theoretical significance since they correspond to physically important situations. However, in spite of developing analytical approaches dedicated to the DPL model, most solutions still available in the literature are obtained by numerical methods, and there are few exact solutions [11,12].…”

Thermal wave crystals based on the dual-phase-lag model

“…Reviews on the DPL theory were given by Ozisik et al [22], Xu et al [23] and Fan et al [24] among others. Zhang et al [25] presented two exact explicit solutions for the three-dimensional DPL heat conduction equation. Chou and Yang [26,27] examined the DPL thermal behavior in 1D and two-dimensional (2D) single-/multi-layered structures.…”